1. Field of the Invention
The invention is applicable to the implementation of digital demodulation of analog frequency modulated signal in which the frequency of the modulating signal is relatively small compared to the sampling frequency. A typical application of this invention is demodulation of the frequency modulated TV sound sampled at high frequency.
2. Description of the Prior Art
An FM modulated signal y(t) can be expressed as:
y(t)=A sin["psgr"(t)]=A sin[2xcfx80fct+xcfx86(t)]
where A is the amplitude of the modulated signal, fc is the carrier frequency and xcfx86(t) is a function of the baseband (modulating) signal m(t) as follows       φ    ⁢          (      t      )        =      k    ⁢                  ∫                  -          ∞                t            ⁢                        m          ⁢                      (            t            )                          ⁢                  ⅆ          t                    
where k is the modulation constant. The instantaneous frequency f(t) of the modulated signal y(t) is a linear function of m(t) as follows       f    ⁢          (      t      )        =                    1                  2          ⁢          π                    ⁢                        ⅆ                      ψ            ⁢                          (              t              )                                                ⅆ          t                      =                  f        c            +                        k                      2            ⁢            π                          ⁢                  m          ⁢                      (            t            )                              
The FM demodulation is to recover m(t) from the FM modulated signal y(t). There are several methods to demodulate an analog FM modulated signal using digital techniques. The arc tangent FM demodulation method (see xe2x80x9cDemodulator of sampled data FM signals from sets of four successive samplesxe2x80x9d, by J. J. Gibson, U.S. Pat. No. 4,547,737, and xe2x80x9cDigital Frequency discriminatorxe2x80x9d, by F. G. A. Goupe, Electronics Letters, vol. 15, no. 16, August 1979) uses the Hilbert transform to decompose the carrier into two signals a(t) and b(t) which are orthogonal to each other, i.e.
y(t)=a(t)+jb(t)
Thus, the baseband signal m(t) can be obtained by       m    ⁢          (      t      )        =      α    ⁢                  ⅆ                  {                                    tan                              -                1                                      ⁡                          [                                                b                  ⁢                                      (                    t                    )                                                  /                                  a                  ⁢                                      (                    t                    )                                                              ]                                }                            ⅆ        t            
where a is a constant. Another method is proposed based on an arc cosine system (see xe2x80x9cDigital FM demodulation apparatus demodulating sampled digital FM modulated wavexe2x80x9d by Y. Ishikawa and S. Nomura, U.S. Pat. No. 5,444,416). In this arc cosine FM demodulation method, the FM modulated signal is first sampled at a sampling interval T. Consecutive samples ytxe2x88x92T, yt and yt+T at time txe2x88x92T, t and t+T can be expressed as follows:             y              t        -        T              =          A      ⁢              xe2x80x83            ⁢              sin        ⁡                  [                                    2              ⁢              π              ⁢                              xe2x80x83                            ⁢                              f                ⁢                                  (                                      t                    -                    T                                    )                                                      +            θ                    ]                                y      t        =          A      ⁢              xe2x80x83            ⁢              sin        ⁡                  [                                    2              ⁢              π              ⁢                              xe2x80x83                            ⁢              f              ⁢                              xe2x80x83                            ⁢              t                        +            θ                    ]                                y              t        -        T              =          A      ⁢              xe2x80x83            ⁢              sin        ⁡                  [                                    2              ⁢              π              ⁢                              xe2x80x83                            ⁢                              f                ⁢                                  (                                      t                    +                    T                                    )                                                      +            θ                    ]                    
where A, f and xcex8 are an amplitude, an instantaneous frequency and an initial phase of yt, respectively. It has been proved that the instantaneous frequency f at time t is given by:   f  =            1              2        ⁢        π        ⁢                  xe2x80x83                ⁢        T              ⁢                  cos                  -          1                    ⁢              (                                            y                              t                -                T                                      +                          y                              t                +                T                                                          2            ⁢                          y              t                                      )            
Once the instantaneous frequency of a FM modulated i signal is obtained, the demodulated signal can be obtained from the instantaneous frequency using a linear equation.
As described above, Hilbert transform is used in the arc tangent digital demodulation method is a phase circuit employing the Hilbert transform. The required circuit is large and the computation process introduces long delay. In order to avoid this problem, the sampling frequency must be carefully chosen to be four times the carrier frequency rate. This is not practical. Firstly, different FM systems use different carrier frequencies. The demodulation system will be too complex if multiple frequencies are introduced. Secondly, the sampling frequency is sometimes restricted by requirements of other modules of an FM demodulation and signal processing system. The arc cosine digital FM demodulation method is simpler than the arc tangent FM digital demodulation system. However, there is still an inconvenient division operation involved in the procedure for computing the instantaneous frequency. This is not practical in hardware implementation since the values of yt are sometimes close or equal to zero. Thus, special processing is needed to prevent the demodulation system from overflow if this situation happens.
The problem to be solved by this invention is to develop a simple method for digital FM demodulation having less complexity than the methods described in the prior art.
The new method of calculating the instantaneous frequency based on the sampled values is described as follows. Assume that an FM modulated signal is sampled at a sampling interval of T. If the sampling frequency is high enough, we can assume that the instantaneous frequency values at time txe2x88x92T, t and t+T are approximately the same, i.e.
ftxe2x88x92T≈ft≈ft+T
Hence, three consecutive samples ytxe2x88x92T, ytxe2x88x92T and ytxe2x88x92T at time txe2x88x92T, t and t+T can be respectively expressed as follows:             y              t        -        T              =          A      ⁢              xe2x80x83            ⁢              sin        ⁡                  [                                    2              ⁢              π              ⁢                              xe2x80x83                            ⁢                                                f                  t                                ⁢                                  (                                      t                    -                    T                                    )                                                      +            θ                    ]                                y      t        =          A      ⁢              xe2x80x83            ⁢              sin        ⁡                  [                                    2              ⁢              π              ⁢                              xe2x80x83                            ⁢                              f                t                            ⁢                              xe2x80x83                            ⁢              t                        +            θ                    ]                                y              t        -        T              =          A      ⁢              xe2x80x83            ⁢              sin        ⁡                  [                                    2              ⁢              π              ⁢                              xe2x80x83                            ⁢                                                f                  t                                ⁢                                  (                                      t                    +                    T                                    )                                                      +            θ                    ]                    
Let xcfx86=2xcfx80ftt+xcex8 and xcfx860=2xcfx80ftT, we have             y              t        -        T              =          A      ⁢              xe2x80x83            ⁢              sin        ⁢                  (                      ϕ            -                          ϕ              0                                )                                y      t        =          A      ⁢              xe2x80x83            ⁢      sin      ⁢              xe2x80x83            ⁢      ϕ                  y              t        -        T              =          A      ⁢              xe2x80x83            ⁢              sin        ⁢                  (                      ϕ            +                          ϕ              0                                )                    
Since             y              t        -        T              ⁢          y              t        +        T              =                    A        2            ⁢              sin        ⁢                  (                      ϕ            -                          ϕ              0                                )                    ⁢              sin        ⁢                  (                      ϕ            +                          ϕ              0                                )                      =                            A          2                2            ⁡              [                              cos            ⁢                          (                              2                ⁢                                  ϕ                  0                                            )                                -                      cos            ⁢                          (                              2                ⁢                ϕ                            )                                      ]            
and       cos    ⁢          (              2        ⁢        ϕ            )        =            1      -              2        ⁢                  sin          2                ⁢        ϕ              =          1      -                        2                      A            2                          ⁢                  y          t          2                    
Therefore       cos    ⁢          (              2        ⁢                  ϕ          0                    )        =                    2                  A          2                    ⁢              (                                            y                              t                -                T                                      ⁢                          y                              t                +                T                                              -                      y            t            2                          )              +    1  
Thus, the instantaneous frequency value at time t can be computed by the following equation:       f    t    =            1              4        ⁢        π        ⁢                  xe2x80x83                ⁢        T              ⁢                  cos                  -          1                    ⁡              [                                            2                              A                2                                      ⁢                          (                                                                    y                                          t                      -                      T                                                        ⁢                                      y                                          t                      +                      T                                                                      -                                  y                  t                  2                                            )                                +          1                ]            
Based on the equations described here, an apparatus is set up for measuring instantaneous frequency of FM-modulated signal. Said apparatus comprises a sampling means, an instantaneous frequency computing means and a lowpass filtering means. Said instantaneous frequency computing means further comprises two delay means, a multiplier, a square means, a bit-shifting means, a subtracting means, a scaling means, an adding means and an inverse cosine computing means.
The description will now be made on the operation of the apparatus invented for digital FM demodulation. Said sampling means samples the input analog FM modulated signal at a prescribed sampling interval T to obtain three consecutive samples Ytxe2x88x92T, yt and yt+T at times txe2x88x92T, t and t+T, respectively. Said signal instantaneous frequency computing means computes said instantaneous frequency ft at time t based on an equation             f      t        =                  1                  4          ⁢          π          ⁢                      xe2x80x83                    ⁢          T                    ⁢                        cos                      -            1                          ⁡                  (                      s            t                    )                      ,
where       s    t    =                    2                  A          2                    ⁢              (                                            y                              t                -                T                                      ⁢                          y                              t                +                t                                              -                      y            t            2                          )              +    1  
and A is the amplitude of an FM modulated signal. The instantaneous frequency is then filtered by said lowpass filter to reduce the noise introduced by said data sampling means and said instantaneous frequency computing means.
The operations of said instantaneous frequency computing means are now explained. The first delay means receives the digitized FM signal and delays said digitized FM signal by one prescribed interval to provide a first delayed FM signal. The second delay means receives said first delayed FM signal and further delays said first delayed FM signal by said prescribed interval to provide a second delayed FM signal. The square means receives said first delayed signal and computes the squared values of said first delayed FM signals. The multiplier computes the product values of digitized FM signal and said second delayed FM signal. Said squared values are subtracted from said product values to provide the difference signal. Each value of said difference signal is left-shifted by one bit to the left by said bit-shifting means to obtain bit-shifted signal. The scaled signal is then obtained by scaling said bit-shifted signal by the factor A2, where A is the amplitude of the FM signal. The adding means increments each value of said scaled signal by one to obtain said incremented signal. The instantaneous frequency is computed by said inverse cosine computing means based on said incremented signal.